Noncommutative Pressure and the Variational Principle in Cuntz–Krieger-type C*-Algebras
✍ Scribed by David Kerr; Claudia Pinzari
- Publisher
- Elsevier Science
- Year
- 2002
- Tongue
- English
- Weight
- 378 KB
- Volume
- 188
- Category
- Article
- ISSN
- 0022-1236
No coin nor oath required. For personal study only.
✦ Synopsis
Let a be a self-adjoint element of an exact C g -algebra A and h: A Q A a contractive completely positive map. We define a notion of dynamical pressure P h (a) which adopts Voiculescu's approximation approach to noncommutative entropy and extends the Voiculescu-Brown topological entropy and Neshveyev and Størmer unital-nuclear pressure. A variational inequality bounding P h (a) below by the free energies h s (h)+s(a) with respect to the Sauvageot-Thouvenot entropy h s (h) is established in two stages via the introduction of a local state approximation entropy, whose associated free energies function as an intermediate term. Pimsner C g -algebras furnish a framework for investigating the variational principle, which asserts the equality of P h (a) with the supremum of the free energies over all h-invariant states. In one direction we extend Brown's result on the constancy of the Voiculescu-Brown entropy upon passing to the crossed product, and in another we show that the pressure of a self-adjoint element over the Markov subshift underlying the canonical map on the Cuntz-Krieger algebra O A is equal to its classical pressure. The latter result is extended to a more general setting comprising an expanded class of Cuntz-Krieger-type Pimsner algebras, leading to the variational principle for self-adjoint elements in a diagonal subalgebra. Equilibrium states are constructed from KMS states under certain conditions in the case of Cuntz-Krieger algebras.